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Properties

Label 4719.2.a.bo
Level $4719$
Weight $2$
Character orbit 4719.a
Self dual yes
Analytic conductor $37.681$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4719,2,Mod(1,4719)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4719, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4719.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4719 = 3 \cdot 11^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4719.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(37.6814047138\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 17 x^{12} + 53 x^{11} + 105 x^{10} - 347 x^{9} - 296 x^{8} + 1059 x^{7} + 404 x^{6} + \cdots + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 429)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{9} + 1) q^{5} + \beta_1 q^{6} - \beta_{5} q^{7} + ( - \beta_{11} + \beta_{9} - \beta_{6} + \cdots - 1) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{9} + 1) q^{5} + \beta_1 q^{6} - \beta_{5} q^{7} + ( - \beta_{11} + \beta_{9} - \beta_{6} + \cdots - 1) q^{8}+ \cdots + ( - 2 \beta_{13} + 3 \beta_{12} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{2} - 14 q^{3} + 15 q^{4} + 8 q^{5} + 3 q^{6} - 3 q^{7} - 9 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 3 q^{2} - 14 q^{3} + 15 q^{4} + 8 q^{5} + 3 q^{6} - 3 q^{7} - 9 q^{8} + 14 q^{9} + q^{10} - 15 q^{12} - 14 q^{13} + 18 q^{14} - 8 q^{15} + 13 q^{16} + q^{17} - 3 q^{18} + 10 q^{19} + 32 q^{20} + 3 q^{21} + 15 q^{23} + 9 q^{24} + 24 q^{25} + 3 q^{26} - 14 q^{27} - 4 q^{28} - 18 q^{29} - q^{30} + 8 q^{31} - 11 q^{32} - 12 q^{34} + 23 q^{35} + 15 q^{36} + q^{37} + 14 q^{38} + 14 q^{39} - 46 q^{40} - 18 q^{42} + 6 q^{43} + 8 q^{45} + 26 q^{47} - 13 q^{48} + 23 q^{49} - 4 q^{50} - q^{51} - 15 q^{52} + 8 q^{53} + 3 q^{54} + 24 q^{56} - 10 q^{57} + 10 q^{59} - 32 q^{60} - 14 q^{61} - 65 q^{62} - 3 q^{63} + 43 q^{64} - 8 q^{65} - 44 q^{67} + 38 q^{68} - 15 q^{69} + 67 q^{70} + 37 q^{71} - 9 q^{72} - 13 q^{73} + 58 q^{74} - 24 q^{75} + 62 q^{76} - 3 q^{78} + 14 q^{79} + 56 q^{80} + 14 q^{81} - 14 q^{82} - 36 q^{83} + 4 q^{84} - 5 q^{85} + 25 q^{86} + 18 q^{87} + 63 q^{89} + q^{90} + 3 q^{91} + 34 q^{92} - 8 q^{93} + 41 q^{94} - 9 q^{95} + 11 q^{96} + 26 q^{97} + 33 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 17 x^{12} + 53 x^{11} + 105 x^{10} - 347 x^{9} - 296 x^{8} + 1059 x^{7} + 404 x^{6} + \cdots + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2405 \nu^{13} - 8446 \nu^{12} + 33447 \nu^{11} + 201319 \nu^{10} - 87265 \nu^{9} - 1723743 \nu^{8} + \cdots - 683619 ) / 135741 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2869 \nu^{13} - 3359 \nu^{12} + 66258 \nu^{11} + 60056 \nu^{10} - 561839 \nu^{9} - 385818 \nu^{8} + \cdots - 33744 ) / 135741 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9071 \nu^{13} - 4990 \nu^{12} + 231585 \nu^{11} + 38905 \nu^{10} - 2160091 \nu^{9} + \cdots + 684384 ) / 135741 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9806 \nu^{13} - 18768 \nu^{12} - 174830 \nu^{11} + 305099 \nu^{10} + 1155204 \nu^{9} + \cdots - 232858 ) / 135741 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11966 \nu^{13} - 17485 \nu^{12} - 212113 \nu^{11} + 260594 \nu^{10} + 1381361 \nu^{9} + \cdots - 31559 ) / 135741 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4582 \nu^{13} + 10643 \nu^{12} + 80806 \nu^{11} - 178959 \nu^{10} - 535006 \nu^{9} + \cdots + 292203 ) / 45247 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14651 \nu^{13} - 12075 \nu^{12} + 331670 \nu^{11} + 255682 \nu^{10} - 2815971 \nu^{9} + \cdots - 246632 ) / 135741 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19115 \nu^{13} - 27504 \nu^{12} - 366491 \nu^{11} + 466787 \nu^{10} + 2616723 \nu^{9} + \cdots - 655546 ) / 135741 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24457 \nu^{13} + 6693 \nu^{12} + 506500 \nu^{11} - 49417 \nu^{10} - 3971175 \nu^{9} + \cdots - 149515 ) / 135741 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 51598 \nu^{13} - 83768 \nu^{12} - 972269 \nu^{11} + 1359466 \nu^{10} + 6931234 \nu^{9} + \cdots - 889273 ) / 135741 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17281 \nu^{13} - 25535 \nu^{12} - 322811 \nu^{11} + 400884 \nu^{10} + 2264114 \nu^{9} - 2175091 \nu^{8} + \cdots + 45337 ) / 45247 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{9} + \beta_{6} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} + \beta_{11} - \beta_{9} + \beta_{7} + \beta_{3} + 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{11} - 9\beta_{9} + 9\beta_{6} + \beta_{5} - 4\beta_{4} + \beta_{3} + \beta_{2} + 29\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9 \beta_{13} - 10 \beta_{12} + 10 \beta_{11} + \beta_{10} - 11 \beta_{9} + \beta_{8} + 13 \beta_{7} + \cdots + 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{13} - 2 \beta_{12} + 66 \beta_{11} - 67 \beta_{9} + \beta_{8} + \beta_{7} + 71 \beta_{6} + \cdots + 98 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 65 \beta_{13} - 78 \beta_{12} + 79 \beta_{11} + 12 \beta_{10} - 91 \beta_{9} + 19 \beta_{8} + 127 \beta_{7} + \cdots + 580 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 13 \beta_{13} - 29 \beta_{12} + 458 \beta_{11} - 4 \beta_{10} - 472 \beta_{9} + 22 \beta_{8} + \cdots + 814 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 443 \beta_{13} - 568 \beta_{12} + 582 \beta_{11} + 101 \beta_{10} - 685 \beta_{9} + 229 \beta_{8} + \cdots + 3986 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 117 \beta_{13} - 296 \beta_{12} + 3132 \beta_{11} - 72 \beta_{10} - 3265 \beta_{9} + 306 \beta_{8} + \cdots + 6541 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2969 \beta_{13} - 4054 \beta_{12} + 4183 \beta_{11} + 733 \beta_{10} - 4971 \beta_{9} + 2292 \beta_{8} + \cdots + 28273 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 917 \beta_{13} - 2658 \beta_{12} + 21405 \beta_{11} - 848 \beta_{10} - 22497 \beta_{9} + 3456 \beta_{8} + \cdots + 51619 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.77176
2.53830
1.90905
1.78171
1.67845
1.29581
0.315345
0.0903027
−0.581165
−0.975154
−1.45902
−1.57932
−2.18492
−2.60115
−2.77176 −1.00000 5.68266 0.894542 2.77176 −1.24009 −10.2074 1.00000 −2.47946
1.2 −2.53830 −1.00000 4.44296 4.23995 2.53830 −2.90987 −6.20095 1.00000 −10.7623
1.3 −1.90905 −1.00000 1.64447 2.25639 1.90905 3.35031 0.678719 1.00000 −4.30756
1.4 −1.78171 −1.00000 1.17448 −2.78091 1.78171 0.261118 1.47084 1.00000 4.95476
1.5 −1.67845 −1.00000 0.817197 −0.775974 1.67845 −0.307069 1.98528 1.00000 1.30243
1.6 −1.29581 −1.00000 −0.320873 −1.33560 1.29581 −5.13358 3.00741 1.00000 1.73068
1.7 −0.315345 −1.00000 −1.90056 −2.66732 0.315345 1.52108 1.23002 1.00000 0.841126
1.8 −0.0903027 −1.00000 −1.99185 3.11967 0.0903027 −2.74320 0.360475 1.00000 −0.281715
1.9 0.581165 −1.00000 −1.66225 −2.99274 −0.581165 −0.398009 −2.12837 1.00000 −1.73927
1.10 0.975154 −1.00000 −1.04907 3.59952 −0.975154 3.45077 −2.97332 1.00000 3.51009
1.11 1.45902 −1.00000 0.128732 0.403770 −1.45902 −4.80858 −2.73021 1.00000 0.589108
1.12 1.57932 −1.00000 0.494242 3.53721 −1.57932 4.79359 −2.37807 1.00000 5.58638
1.13 2.18492 −1.00000 2.77387 −1.80490 −2.18492 −0.930072 1.69085 1.00000 −3.94356
1.14 2.60115 −1.00000 4.76599 2.30638 −2.60115 2.09360 7.19477 1.00000 5.99924
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4719.2.a.bo 14
11.b odd 2 1 4719.2.a.bp 14
11.d odd 10 2 429.2.n.c ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.n.c ✓ 28 11.d odd 10 2
4719.2.a.bo 14 1.a even 1 1 trivial
4719.2.a.bp 14 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4719))\):

\( T_{2}^{14} + 3 T_{2}^{13} - 17 T_{2}^{12} - 53 T_{2}^{11} + 105 T_{2}^{10} + 347 T_{2}^{9} - 296 T_{2}^{8} + \cdots + 11 \) Copy content Toggle raw display
\( T_{5}^{14} - 8 T_{5}^{13} - 15 T_{5}^{12} + 242 T_{5}^{11} - 90 T_{5}^{10} - 2841 T_{5}^{9} + \cdots + 13145 \) Copy content Toggle raw display
\( T_{7}^{14} + 3 T_{7}^{13} - 56 T_{7}^{12} - 146 T_{7}^{11} + 1109 T_{7}^{10} + 2451 T_{7}^{9} + \cdots + 1280 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 3 T^{13} + \cdots + 11 \) Copy content Toggle raw display
$3$ \( (T + 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} - 8 T^{13} + \cdots + 13145 \) Copy content Toggle raw display
$7$ \( T^{14} + 3 T^{13} + \cdots + 1280 \) Copy content Toggle raw display
$11$ \( T^{14} \) Copy content Toggle raw display
$13$ \( (T + 1)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} - T^{13} + \cdots - 12766976 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 295096320 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 163321600 \) Copy content Toggle raw display
$29$ \( T^{14} + 18 T^{13} + \cdots + 3480320 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 5841929984 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 6838479616 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 21028423424 \) Copy content Toggle raw display
$43$ \( T^{14} - 6 T^{13} + \cdots - 33677411 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 1196470755 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 2986562304 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 7628417695 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 258113691 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 3839792384 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 1636322382080 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 336253696 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 1112616355 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 4833701501211 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 4529834245376 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 8842533120 \) Copy content Toggle raw display
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